1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Nesterboy [21]
3 years ago
13

1) A thin ring made of uniformly charged insulating material has total charge Q and radius R. The ring is positioned along the x

-y plane of a 3d coordinate system such that the center of the ring is at the origin of the coordinate system. (a) Determine an expression for the potential at an arbitrary location along the z-axis in terms of Q, R, and z. (b) Use this expression to determine an expression for the magnitude of the electric field at an arbitrary location along the z-axis in terms of Q, R, and z. Hint: Apply the technique of charge integration in part (a) and poten
Physics
1 answer:
allochka39001 [22]3 years ago
4 0

Answer:

(A) considering the charge "q" evenly distributed, applying the technique of charge integration for finite charges, you obtain the expression for the potential along any point in the Z-axis:

V(z)=\frac{Q}{4\pi (\epsilon_{0}) \sqrt{R^{2} +z^{2}}  }

With (\epsilon_{0}) been the vacuum permittivity

(B) The expression for the magnitude of the E(z) electric field along the Z-axis is:

E(z)=\frac{QZ}{4\pi (\epsilon_{0}) (R^{2} +z^{2})^{\frac{3}{2} }    }

Explanation:

(A) Considering a uniform linear density λ_{0} on the ring, then:

dQ=\lambda dl (1)⇒Q=\lambda_{0} 2\pi R(2)⇒\lambda_{0}=\frac{Q}{2\pi R}(3)

Applying the technique of charge integration for finite charges:

V(z)= 4\pi (ε_{0})\int\limits^a_b {\frac{1}{ r'  }} \, dQ(4)

Been r' the distance between the charge and the observation point and a, b limits of integration of the charge. In this case a=2π and b=0.

Using cylindrical coordinates, the distance between a point of the Z-axis and a point of a ring with R radius is:

r'=\sqrt{R^{2} +Z^{2}}(5)

Using the expressions (1),(4) and (5) you obtain:

V(z)= 4\pi (\epsilon_{0})\int\limits^a_b {\frac{\lambda_{0}R}{ \sqrt{R^{2} +Z^{2}}  }} \, d\phi

Integrating results:

V(z)=\frac{Q}{4\pi (\epsilon_{0}) \sqrt{R^{2} +z^{2}}  }   (S_a)

(B) For the expression of the magnitude of the field E(z), is important to remember:

|E| =-\nabla V (6)

But in this case you only work in the z variable, soo the expression (6) can be rewritten as:

|E| =-\frac{dV(z)}{dz} (7)

Using expression (7) and (S_a), you get the expression of the magnitude of the field E(z):

E(z)=\frac{QZ}{4\pi (\epsilon_{0}) (R^{2} +z^{2})^{\frac{3}{2} }    } (S_b)

You might be interested in
What can you infer from the fact that metals are good conductors of electricity?
LuckyWell [14K]

Answer:

Knowing that these metals are infact good conductors of electricity we can infer that metals are able to hold and conduct certain temperatures. Another thing we can infer is that these good conductors can be used in connection to transferring energy or electricity.

6 0
3 years ago
Read 2 more answers
A 0.00275 kg air‑inflated balloon is given an excess negative charge q1 =−3.50×10−8 C by rubbing it with a blanket. It is found
kati45 [8]

Answer:

1)  \rm q_2 is<u> positive.</u>

<u></u>

2) \rm q_2=4.56\times 10^{-10}\ C.

Explanation:

<h2><u>Part 1:</u></h2>

<u></u>

The charged rod is held above the balloon and the weight of the balloon acts in downwards direction. To balance the weight of the balloon, the force on the balloon due to the rod must be directed along the upwards direction, which is only possible when the rod exerts an attractive force on the balloon and the electrostatic force on the balloon due to the rod is attractive when the polarities of the charge on the two are different.

Thus, In order for this to occur, the polarity of charge on the rod must be positive, i.e., \rm q_2 is <u>positive.</u>

<u></u>

<h2><u>Part 2:</u></h2>

<u></u>

<u>Given:</u>

  • Mass of the balloon, m = 0.00275 kg.
  • Charge on the balloon, \rm q_1 = -3.50\times 10^{-8}\ C.
  • Distance between the rod and the balloon, d = 0.0640 m.
  • Acceleration due to gravity, \rm g = 9.81\ m/s^2.

In order to balloon to be float in air, the weight of the balloom must be balanced with the electrostatic force on the balloon due to rod.

Weight of the balloon, \rm W = mg = 0.00275\times 9.81=2.70\times 10^{-2}\ N.

The magnitude of the electrostatic force on the balloon due to the rod is given by

\rm F_e = \dfrac{1}{4\pi \epsilon_o}\dfrac{|q_1||q_2|}{d^2}.

\rm \dfrac{1}{4\pi \epsilon_o} is the Coulomb's constant.

For the elecric force and the weight to be balanced,

\rm F_e = W\\\dfrac{1}{4\pi \epsilon_o}\dfrac{|q_1||q_2|}{d^2}=W\\8.99\times 10^9\times \dfrac{3.50\times10^{-8}\times |q_2| }{0.0640^2}=2.70\times 10^{-2}\\|q_2| = \dfrac{2.70\times 10^{-2}\times 0.00640^2}{8.99\times 10^9\times 2.70\times 10^{-7}}=4.56\times 10^{-10}\ C.

3 0
3 years ago
What is the soil fabric?
inessss [21]

Answer:

The term soil fabric refers to the geometrical arrangement of individual particles in a soil, including the geometrical distribution of pore spaces. Soil fabric may be used to describe the particle arrangement in both cohesive soils such as clays and granular soils such as silts, sands, and gravels.

5 0
2 years ago
A bike rider pedals with constant acceleration to reach a velocity of 7.8 m/s over a time of 4.2 s. during the period of acceler
Artyom0805 [142]

To calculate the initial velocity of the bike, we use the following equation

d=\frac{1}{2} (u+v)t.

or

u=\frac{2d}{t} -v

Here, u is initial velocity, v is final velocity, t is the time and d is the distance covered by bike.

Given, u =7.8 m/s,d= 19 m and t=4.2 s.

Substituting these values in above equation, we get

u = \frac{2 \times 19}{4.2 \ s} -7.8 m/s = 9.05 \ m/s - 7.8 \ m/s \\\\ u= 1.2 m/s.

Thus, the initial velocity of the bike is 1.2 m/s.

3 0
3 years ago
Define fundamental unit ? write its two example. ​
stealth61 [152]

the International System of Units, the fundamental units are: The meter (symbol: m), used to measure length. The kilogram (symbol: kg), used to measure mass. The second (symbol: s), used to measure time. The ampere (symbol: A), used to measure electric current.

8 0
3 years ago
Other questions:
  • It takes you 7.7 min to walk with an average velocity of 1.7 m/s to the north from the bus stop to the museum entrance. How far
    15·1 answer
  • A battery is rated at 12 V and 160 A-h. How much energy does the battery store? What is the cost of this energy at $0. 15/kWh?
    7·1 answer
  • What is one key physical difference between transition metals and poor metals?
    12·1 answer
  • Bill is throwing a football at four targets and attempting to knock them over. Which of the following targets will be hardest fo
    6·1 answer
  • A ball is thrown up with a speed of 30 m/s and caught when it comes back down. You can ignore air resistance.
    9·1 answer
  • Which of the following is an example of technology influencing science?
    15·2 answers
  • If you were on a ship at sea, and a tsunami passed under your ship, what would probably be your reaction? explain.
    15·1 answer
  • What is the average speed of a race car that moved 20 kilometers in 10 minutes?
    7·1 answer
  • If a car has a velocity of 85 km/hr, how long will it take to accelerate to 45 km/hr if the acceleration is -3 km/hr/sec?
    10·1 answer
  • Help me with this please
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!